[Math] Prove that the two integrals are equal with periodic function

Question

Problem: Let f be a real-valued continuous periodic function with period T. Prove that $$\int_{a}^{a+T}f(x)dx = \int_{0}^{T}f(x)dx$$

Any good hints? My following strategy is to try to show that the Riemann integrals are equal by showing that the two partitions for each integrals have equal length $\frac{T}{n}$, and the value $f(x_{k})$ in one partition also exist in another partition for the other integral.

Answer 1

Define $g(a) = \int_a^{a+T} f(x) \, dx$. Then $g$ is differentiable and $$g'(a) = f(a+T) - f(a) = 0,$$ so $g$ is constant.